Intro

Issue

TROLL model currently compute leaf lifespan with Reich’s allometry (Reich et al. 1991). But we have shown that Reich’s allometry is underestimating leaf lifespan for low LMA species. Moreover simulations estimated unrealistically low aboveground biomass for low LMA species. We assumed Reich’s allometry underestimation of leaf lifespan for low LMA species being the source of unrealistically low aboveground biomass inside TROLL simulations. We decided to find a better allometry with Wright et al. (2004) GLOPNET dataset.

We tested different models starting from complet model Mcomp (with logged mean): \[ {LL_s}_j \sim \mathcal{logN}({\beta_0} + {\beta_1}_s*{LMA_s}_j^{{\beta_3}_s} - {\beta_2}_s*{Nmass_s}_j^{{\beta_4}_s},\,\sigma)\,\]

\[s=1,...,S_{=4}~, ~~j=1,...,n_s\] \[{\beta_i}_s \sim \mathcal{N}({\beta_i},\,\sigma_i)\,^I, ~~(\beta_i, \sigma, \sigma_i) \sim \mathcal{\Gamma}(0.001,\,0.001)\,^{2I+1}\] We tested models M1 to M9 detailed in following tabs to find the better trade-off between:

  1. Complexity (and number of parameters)
  2. Convergence
  3. Likelihood
  4. Prediction quality with Root Mean Square Error of Prediction (RMSEP)

RMSEP was computed by fitting a model without a dataset and evaluating it with the same dataset. RMSEP in results are mean RMSEP for each dataset. Results are shown for each models in eahc model tabs and summarized in Results tab.

Table 2: Models summary.
M Model
M1 \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - {\beta_2}_s*N,\sigma)\,\)
M2 \(LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\)
M3 \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - {\beta_2}_s*N^{{\beta_4}_s},\sigma)\,\)
M4 \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - N,\sigma)\,\)
M5 \(LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\)
M6 \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\)
M7 \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA,\sigma)\,\)
M8 \(LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s},\sigma)\,\)
M9 \(LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s},\sigma)\,\)

LL graph

Figure 1: Leaf mass per area (LMA), leaf nitrogen content (Nmass) and leaflifespan (LL). Leaf mass per area (LMA in \(g.m^{-2}\)), leaf nitrogen content (Nmass, in \(mg.g^-1\)) and leaf lifespan (LL in \(months\)) are taken in GLOPNET dataset from Wright et al. (2004).

M1

Model

\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - {\beta_2}_s*N,\sigma)\,\] Maximum likekihood of 11.1880728 and RMSEP of 12.346

Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.

Convergence

Parameters markov chains. Light grey area represents warmup iterations.

Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.

M2

Model

\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of -0.5940783 and RMSEP of 11.402

Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.

Convergence

Parameters markov chains. Light grey area represents warmup iterations.

Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.

M3

Model

\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - {\beta_2}_s*N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of 16.8245623 and RMSEP of 15.029

Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.

Convergence

Parameters markov chains. Light grey area represents warmup iterations.

Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.

M4

Model

\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA - N,\sigma)\,\] Maximum likekihood of 4.9012658 and RMSEP of 14.26

Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.

Convergence

Parameters markov chains. Light grey area represents warmup iterations.

Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.

M5

Model

\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of -1.2010255 and RMSEP of 12.077

Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.

Convergence

Parameters markov chains. Light grey area represents warmup iterations.

Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.

M6

Model

\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s} - N^{{\beta_4}_s},\sigma)\,\] Maximum likekihood of 15.1501017 and RMSEP of 13.059

Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.

Convergence

Parameters markov chains. Light grey area represents warmup iterations.

Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.

M7

Model

\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA,\sigma)\,\] Maximum likekihood of 4.1744354 and RMSEP of 36.089

Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.

Convergence

Parameters markov chains. Light grey area represents warmup iterations.

Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.

M8

Model

\[ LL \sim \mathcal{logN}(\beta_0 + LMA^{{\beta_3}_s},\sigma)\,\] Maximum likekihood of -8.1187308 and RMSEP of 11.919

Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.

Convergence

Parameters markov chains. Light grey area represents warmup iterations.

Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.

M9

Model

\[ LL \sim \mathcal{logN}(\beta_0 + {\beta_1}_s*LMA^{{\beta_3}_s},\sigma)\,\] Maximum likekihood of 9.4315134 and RMSEP of 17.553

Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.

Convergence

Parameters markov chains. Light grey area represents warmup iterations.

Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.

Results

Column

Model M1 seemed to have shown the best trade-off between complexity (\(K=6\) parameters), convergence (see tab M1 ), likelihood, and prediction quality (see table 2). Figure 2 presents model prediction confidence interval and figure 3 compare Reich’s allometry and model M1 predictions with species functional traits used in TROLL. Nevertheless we will test other predictors gathered from TRY database following a model with the same response curve (see leaf lifespan ).

Table 2: Models likelihood and prediction quality.Root mean square errof of prediction (RMSEP) was computed by fitting a model without a dataset and evaluating it with the same dataset. RMSEP in results are mean RMSEP for each dataset.

ML RMSEP
M1 11.188 12.34647
M2 -0.594 11.40243
M3 16.825 15.02891
M4 4.901 14.26035
M5 -1.201 12.07721
M6 15.150 13.05924
M7 4.174 36.08934
M8 -8.119 11.91863
M9 9.432 17.55283

\[LL = 11.357*e^{0.008 *LMA -0.419*Nmass }\]

Column

Figure 2: Leaflifespan predictions for model M1 with leaf mass per area (LMA), and leaf nitrogen content (Nmass). Leaf lifespan (LL in \(months\)) is predicted with model M1 fit. Leaf mass per area (LMA in \(g.m^{-2}\)) and leaf nitrogen content (Nmass, in %) are taken in GLOPNET dataset from Wright et al. (2004). Warning Nmass (resp. LMA) is not constant and depend on the closest point value for left (resp. right) graph.

Figure 3: Leaflifespan predictions for model M1 and Reich’s allometry with leaf mass per area (LMA), and leaf nitrogen content (Nmass). Leaf lifespan (LL in \(months\)) is predicted with model M1 fit or Reich’s allometry (Reich et al. 1991). Leaf mass per area (LMA in \(g.m^{-2}\)) and leaf nitrogen content (Nmass, in %) are taken in BRIDGE dataset used by TROLL (Maréchaux & Chave).

References

Maréchaux, I. & Chave, J. Joint simulation of carbon and tree diversity in an Amazonian forest with an individual-based forest model. Inprep, 1–13.

Reich, P.B., Uhl, C., Walters, M.B. & Ellsworth, D.S. (1991). Leaf lifespan as a determinant of leaf structure and function among 23 amazonian tree species. Oecologia, 86, 16–24.

Wright, I.J., Reich, P.B., Westoby, M., Ackerly, D.D., Baruch, Z., Bongers, F., Cavender-Bares, J., Chapin, T., Cornelissen, J.H.C., Diemer, M. & Others. (2004). The worldwide leaf economics spectrum. Nature, 428, 821–827.